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NONNEUTRAL PLASMAS 50 can be followed for 104 or 105 orbits before merger occurs. In the case of classical fluids such as water, merger or dissipation typically occurs in a few orbits. Finally, a nonneutral plasma exhibits a wide range of collective waves and instabilities analogous to those observed in an electrically neutral plasma, appropriately modified by self-field effects. These collective waves and instabilities have been documented extensively in theoretical analyses, and algorithms have been developed for calculating the detailed stability behavior or nonneutral plasma over a wide range of system parameters. In addition, a kinetic stability theorem has been developed that determines a sufficient condition for the nonlinear stability of cylindrically symmetric equilibria to arbitrary-amplitude perturbations, including the influence of strong self-field effects. Ion Plasmas Another type of single-component plasma that has been studied extensively is the magnetized, pure ion plasma, confined in a Penning trap, and cooled by laser radiation. In this case, the laser light is used both to cool the ions and to exert a torque on the plasma. This torque has the effect of spinning up the plasma and compressing it. An analytical theory of the collective modes of oscillation in these plasmas has been formulated on the basis of cold fluid theory. This is the first analytical description of the modes of a magnetized, three-dimensional plasma of finite extent with realistic boundary conditions. There is good agreement between theory and the experimental observation of these modes. One result of this increased understanding is that these modes can now be used for the manipulation and confinement of pure ion plasmas. An exact nonlinear theory has been developed for the case of large-amplitude quadrupole modes of oscillation of these plasmas. In ion plasmas that are laser-cooled to cryogenic temperatures, the average kinetic energy per particle can be made small compared to the average interaction potential energy. (These plasmas are often referred to as strongly coupled plasmas.) The resulting ion clouds can form the analogues of dense liquid and solid phases. (See Figure 2.1.) Theoretical and experimental progress has been made recently in understanding the ordering and equilibrium states of these systems. As the temperature is lowered, theory predicts that the ions arrange themselves in concentric spheroidal shells. These shells are the analogues of crystal planes, except that the planes are deformed into spheroids because of the small plasma size. This shell structure has been observed experimentally, with optical imaging techniques, for plasmas up to about 15 shells. Theory predicts that the sample must contain about 60 shells to result in the structure predicted for plasmas of infinite extent (a body-centered-cubic lattice), but this has not yet been tested experimentally. Small numbers of ions have also been confined and cooled in Paul traps,
